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Astronomical Data Analysis Software and Systems VII
ASP Conference Series, Vol. 145, 1998
R. Albrecht, R. N. Hook and H. A. Bushouse, e
Ö Copyright 1998 Astronomical Society of the Pacific. All rights reserved.
ds.
Modelling Spectro­photometric Characteristics of
Nonradially Pulsating Stars
Jadwiga Daszynska and Henryk Cugier
Astronomical Institute of the Wroclaw University PL­51­622 Wroclaw,
ul. Kopernika 11, Poland
Abstract.
We present a computing code for modelling energy flux distributions,
photometric indices and spectral line profiles of (non­)radially pulsating
Main­Sequence stars. The model is based on the perturbation­expansion
formalism taking into account geometrical and nonadiabatic e#ects.
1. Introduction
For a given mode of oscillation the harmonic time dependence, exp(i# nlm t), and
spherical harmonic horizontal dependence, Y m
l
(#, #), are assumed for the first
order perturbed quantities. The mass displacement for the spheroidal modes is
described by y­ and z­eigenfunctions and for toroidal modes by #­eigenfunctions,
cf. Dziembowski & Goode (1992). In the case of slowly rotating stars one
can use the zero­rotation approximation to describe stellar pulsations. Such a
model was used already by Cugier, Dziembowski & Pamyatnykh (1994) to study
nonadiabatic observables of # Cephei stars. Apart from y nlm (r) and z nlm (r) it
is desirable to use the eigenfunction p nlm (r), connected with the Lagrangian
perturbation of pressure, and the f nlm (r)-eigenfunction, which describes the
variations of the local luminosity. In the nonadiabatic theory of pulsation the
eigenvalues # nlm and the eigenfunctions are complex (cf. e.g., Dziembowski
1977) and # nlm = arg(f nlm /y nlm ) is the phase lag between the light and radius
variations.
2. Continuum Flux Behaviour
2.1. Numerical Integration
The monochromatic flux of radiation is given by
F # = # I # (r, #, #,#o · #n)#o · #n
dS
R 2 . (1)
where #o · #n is the scalar product of the observer's direction, #o, and the normal
vector, #n, and dS­ the area of the surface element.
In the program the specific intensity data for the new generation line­
blanketed model atmospheres of Kurucz (1996) were used in order to study the
11

12 Daszynska and Cugier
continuum flux behaviour and photometric indices. Kurucz's (1994) data con­
tain monochromatic fluxes for 1221 wavelengths and monochromatic intensities
at 17 points of ”
µ = #o · #n. Using these data one can interpolate the monochro­
matic intensities for the local values of T e# , log g and ”
µ. We can also introduce
the linear or quadratic shape for the limb­darkening law as defined by Wade &
Rucinski (1985).
2.2. Semi­analytical Method
Integrating Eq.1 over the surface in the linear approach we can obtain the semi­
analytical solution, cf. Daszynska & Cugier (1997) for details,
#F #
F 0
#
= #d lm0 N 0
l # (T 1 +T 2 ) cos((# nlm -m# t+ ”
# nlm )+(T 3 +T 4 +T 5 ) cos(# nlm -m# t # .
(2)
In this formula the temperature e#ects are described by two terms T 1 and T 2 ,
whereas the e#ects of the pressure changes during the pulsation cycle are in­
cluded in T 4 and T 5 . The T 2 and T 5 terms reflect the sensitivity of the limb­
darkening parameters to temperature and gravity variations, respectively. T 3
corresponds to the geometrical e#ects. N 0
l
is a normalizing factor.
3. Accuracy of the Model Calculations
We examined how the results are influenced by di#erent methods of integration
over the stellar surface. The following cases were considered:
­ Model 1: the semi­analytical method (Eq.2) with the quadratic form for
the limb­darkening law,
­ Model 2: the numerical integration of Eq.1 with the quadratic form for
the limb­darkening law; constant limb­darkening coe#cients corresponding to
the equilibrium model were assumed,
­ Model 3: the same as Model 2, but the limb­darkening coe#cients were
interpolated for local values of T e# and log g,
­ Model 4: numerical integration over stellar surface with specific intensities
interpolated for the local values of T e# , log g and ”
µ.
As an example we consider the energy flux distribution and nonadiabatic
observables for a # Cep model. We chose the stellar model (log T 0
e# = 4.33668,
log g 0 = 4.07842) calculated with OPAL opacities. This model shows unstable
l = 0, 1 and 2 modes of oscillations. We calculated theoretical fluxes and the
corresponding Str˜omgren photometric indices at pulsating phases # = 0.05 n
(n=0,...,20). Subsequently amplitudes and phases of the light curves were com­
puted by the least­square method. The accuracy of these calculations can be
estimated from Table 1, which gives the results for the Models 1 ­ 4. The cal­
culations were made on Sun Ultra 1 (192 MB RAM, 166 MHz) computer. The
CPU time per 1 pulsating stellar model is from about 2 seconds (for Model 1)
to about 10 hours (for Model 4).

Modelling Nonradially Pulsating Stars 13
Table 1. Nonadiabatic observables.
l Ay # #y Au/Ay #u - #y Au-y
Ay #u-y - #y
Model 1 0 0.0211 3.3166 2.0024 ­0.0381 0.8241 ­0.0701
Model 2 0 0.0211 3.3167 2.0000 ­0.0381 0.8220 ­0.0715
Model 3 0 0.0213 3.3167 2.0000 ­0.0381 0.8220 ­0.0715
Model 4 0 0.0211 3.3168 2.0000 ­0.0381 0.8217 ­0.0718
Model 1 1 0.0207 3.1916 1.5958 0.0004 0.4876 0.0038
Model 2 1 0.0268 3.1929 1.6119 0.0002 0.4975 0.0022
Model 3 1 0.0268 3.1929 1.6112 0.0002 0.4975 0.0022
Model 4 1 0.0222 3.1910 1.5526 0.0009 0.4535 0.0048
Model 1 2 0.0204 3.2077 1.3476 0.0164 0.2560 0.0790
Model 2 2 0.0195 3.2083 1.3457 0.0161 0.2568 0.0801
Model 3 2 0.0195 3.2083 1.3457 0.0161 0.2568 0.0801
Model 4 2 0.0077 3.2084 1.3170 0.0160 0.2459 0.0804
# assumed
4. Line Profiles
The velocity field of pulsating stars may be found by calculating the time deriva­
tive of the Lagrangian displacement. Including the first order e#ect, the radial
component v p as seen by a distant observer is:
v p = #v puls · (-e z ) = Re{i# nlm [cos ##r(R, #, #, t) - r sin ###(R, #, #, t)]}
= # nlm # cos #r[y nlm (r)+
2m# # 0
nlm
”
y nlm (r)]
l
#
k=-l
d lmk (i)N k
l P k
l (#) sin((# nlm -m# t+k#)
-r sin # # [z nlm (r) +
2m# # 0
nlm
”
z nlm ]
l
#
k=-l
d lmk (i)N k
l
#P k
l
(#)
##
sin((# nlm -m# t + k#)
+ # # l+1,m
sin #
l+1
#
k=-(l+1)
d l+1,m,k (i)kN k
l+1 P k
l+1 (#, #) cos((# nlm - m# t + k#)
+ # # l-1,m
sin #
l-1
#
k=-(l-1)
d l-1,m,k (i)kN k
l-1 P k
l-1 (#, #) cos((# nlm -m# t + k#) ## . (3)
The radial velocity due to pulsation and rotation is then
v r = v p - v e sin i sin # sin #, (4)
where v e corresponds to the equatorial velocity of rotation and i is the angle
between the rotation axis and the direction to the observer.
We illustrate the predicted behaviour of Si III 455.262 nm line profiles for
stellar model given in Sect.3. We considered Kurucz's (1994) model atmospheres

14 Daszynska and Cugier
455.15 455.20 455.25 455.30 455.35
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
l=0
a)
f=0.0
f=0.95
F l
/10
­9
+
Offset
[erg
cm
­2
s
­1
nm
­1 ]
l [nm] 455.15 455.20 455.25 455.30 455.35
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
l=1 m=0
b)
l [nm] 455.15 455.20 455.25 455.30 455.35
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
l=1 m=­1
c)
[nm]
f)
455.15 455.20 455.25 455.30 455.35
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
l=2 m=0
d)
f=0.0
f=0.95
F l
/10
­9
+
Offset
[erg
cm
­2
s
­1
nm
­1 ]
l [nm] 455.15 455.20 455.25 455.30 455.35
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
l=2 m=­1
e)
l [nm] 455.15 455.20 455.25 455.30 455.35
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
l=2 m=­2
f)
l [nm]
f)
Figure 1. Theoretical SiIII 455.262 nm line profiles for various modes.
with the solar chemical composition and the microturbulent velocity v t = 0. All
calculations were made for the amplitude of the stellar radius variations # = 0.01
and rigid rotation. Figures 1a - f show the theoretical line profiles for di#erent
phases of pulsation for i = 77 # and the equatorial velocity V e = 25 km s -1 . The
spectra are given in absolute units. In order to avoid overlap, vertical o#sets
were added to each spectrum using the relationship: F # + n · 0.02 · 10 -9 .
Acknowledgments. This work was supported by the research grant No.2
P03D00108 from the Polish Scientific Research Committee (KBN).
References
Cugier, H., Dziembowski W. A., & Pamyatnykh A. A. 1994, A&A, 291, 143
Daszynska J., & Cugier H. 1997. submitted for publication
Dziembowski W. A. 1977, Acta Astron. 27, 95
Dziembowski W. A., & Goode P. R. 1992, ApJ, 394, 670
Kurucz R. L. 1994, CD­ROM No.19
Kurucz R. L. 1996, private communication
Wade R. A., & Rucinski S. M. 1985, A&AS, 60, 471